I agree with Harvey, but I'd make it stronger: If you value these clippings you definitely ought to print out the scans in addition to keeping the digital copies. Preserving these items with through digital files alone requires relying on a technology that is, as of now, still fickle and fragile. Print copies are another backup.
Print out the clippings on acid and lignin-free paper. If you use good-quality toner and keep the printouts in acid and lignin -free folders in acid free boxes, and away from light they can last for up to 100 years. If you can afford pigment-based ink or toner (depending on whether you have a laser or ink jet printer) the printouts will be more stable and last even longer. None of this is cheap, of course, but it's not impossibly expensive, either. But if you want to be certain to preserve these items, I counsel this belt and suspenders approach. Also extremely important is that you make certain either that the newspaper title and date and page number of the story are visible in each of the scans (it's not good enough to include this information in the file name -- it needs to be actually visible as part of the scan itself), or if you didn't do that at the time of scanning, if you have this information write that it on the back of each printout of a clipping -- in pencil, not pen (unless the ink in the pen is acid-free). If I seem a little fanatical about this, I am, based on more than 25 years as a visual materials archivist.
If you follow the protocols above you can discard the original crumbling clippings in good conscience, although you might want to retain a few of them to have a sample of the flavor of the original object (in which case, store them in clear polyester or polyproplylene sleeves -- these are acid-free and since they are transparent you can see the items immediately).
Good for you for taking the trouble and time to scan these clippings, and for asking how to take care of them properly.
Teaneck, New Jersey